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Closed form expressions for Bayesian sample size. (English) Zbl 1113.62029
Summary: Sample size criteria are often expressed in terms of the concentration of the posterior density, as controlled by some sort of error bound. Since this is done pre-experimentally, one can regard the posterior density as a function of the data. Thus, when a sample size criterion is formalized in terms of a functional of the posterior, its value is a random variable. Generally, such functionals have means under the true distribution.
We give asymptotic expressions for the expected value, under a fixed parameter, for certain types of functionals of the posterior density in a Bayesian analysis. The generality of our treatment permits us to choose functionals that encapsulate a variety of inference criteria and large ranges of error bounds. Consequently, we get simple inequalities which can be solved to give minimal sample sizes needed for various estimation goals. In several parametric examples, we verify that our asymptotic bounds give good approximations to the expected values of the functionals they approximate. Also, our numerical computations suggest our treatment gives reasonable results.

MSC:
 62F15 Bayesian inference 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics
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References:
 [1] Bernardo, J. M. (1997). Statistical inference as a decision problem: The choice of sample size. The Statistician 46 151–153. [2] Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions . Krieger, Melbourne, FL. · Zbl 0657.41001 [3] Cheng, Y., Su, F. and Berry, D. (2003). Choosing sample size for a clinical trial using decision analysis. Biometrika 90 923–936. · Zbl 1436.62523 [4] Clarke, B. S. (1996). Implications of reference priors for prior information and for sample size. J. Amer. Statist. Assoc. 91 173–184. JSTOR: · Zbl 0870.62007 [5] Clarke, B. S. and Sun, D. (1999). Asymptotics of the expected posterior. Ann. Inst. Statist. Math. 51 163–185. · Zbl 0948.62014 [6] DasGupta, A. and Vidakovic, B. (1997). Sample size problems in ANOVA: Bayesian point of view. J. Statist. Plann. Inference 65 335–347. · Zbl 0907.62030 [7] De Santis, F. (2004). Statistical evidence and sample size determination for Bayesian hypothesis testing. J. Statist. Plann. Inference 124 121–144. · Zbl 1094.62032 [8] De Santis, F. (2006). Sample size determination for robust Bayesian analysis. J. Amer. Statist. Assoc. 101 278–291. · Zbl 1118.62314 [9] Goldstein, M. (1981). A Bayesian criterion for sample size. Ann. Statist. 9 670–672. · Zbl 0477.62017 [10] Johnson, R. A. (1967). An asymptotic expansion for posterior distributions. Ann. Math. Statist. 38 1899–1906. · Zbl 0157.46802 [11] Johnson, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Statist. 41 851–864. · Zbl 0204.53002 [12] Joseph, L. and Bélisle, P. (1997). Bayesian sample size determination for normal means and differences between normal means. The Statistician 46 209–226. [13] Joseph, L., du Berger, R. and Bélisle, P. (1997). Bayesian and mixed Bayesian/likelihood criteria for sample size determination. Statistics in Medicine 16 769–781. [14] Joseph, L. and Wolfson, D. (1997). Interval-based versus decision theoretic criteria for the choice of sample size. The Statistician 46 145–149. [15] Katsis, A. and Toman, B. (1999). Bayesian sample size calculations for binomial experiments. J. Statist. Plann. Inference 81 349–362. · Zbl 1057.62507 [16] Lindley, D. (1997). The choice of sample size. The Statistician 46 129–138. [17] Pham-Gia, T. (1997). On Bayesian analysis, Bayesian decision theory and the sample size problem. The Statistician 46 139–144. [18] Pham-Gia, T. and Turkkan, N. (1992). Sample size determination in Bayesian analysis (with discussion). The Statistician 41 389–404. [19] Pham-Gia, T. and Turkkan, N. (2003). Determination of exact sample sizes in the Bayesian estimation of the difference of two proportions. The Statistician 52 131–150. JSTOR: [20] Raiffa, H. and Schlaifer, R. (1961). Applied Statistical Decision Theory . Grad. School Bus. Admin., Harvard Univ., Boston. · Zbl 0952.62008 [21] Sahu, S. K. and Smith, T. M. F. (2004). A Bayesian sample size determination method with practical applications. Available at www.maths.soton.ac.uk/staff/Sahu/research/papers/ssd.pdf. [22] Strasser, H. (1981). Consistency of maximum likelihood and Bayes estimates. Ann. Statist. 9 1107–1113. · Zbl 0483.62019 [23] Wang, F. and Gelfand, A. E. (2002). A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models. Statist. Sci. 17 193–208. · Zbl 1013.62025 [24] Yuan, A. and Clarke, B. (2004). Asymptotic normality of the posterior given a statistic. Canad. J. Statist. 32 119–137. JSTOR: · Zbl 1056.62019
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